CAMOSUN COLLEGE. School of Access. Academic and Career Foundations Department. MATH 052 unit 5. Trigonometry. adapted from:

Transcription

1 CAMOSUN COLLEGE School of Access Academic and Career Foundations Department MATH 052 unit 5 Trigonometry adapted from: ABE Intermediate Level Mathematics Module 14: Trigonometry 1

2 Canadian Cataloguing in Publication Data Main entry under title: Intermediate level mathematics (Adult basic education) ISBN (set) 1. Mathematics Study and teaching (Continuing education) British Columbia. I. Brodie, Sarah. II. British Columbia. Ministry of Advanced Education, Training and Technology. III. Centre for Curriculum, Transfer and Technology. IV. Series. QA14.C32B C Program Information Coordinator, Adult Basic Education Programs Ministry of Advanced Education, Training and Technology Post Secondary Education Division PO Box 9877, Stn Prov Govt Victoria, British Columbia Canada V8W 9T6 Web: Curriculum Information Curriculum Development Team Centre for Curriculum, Transfer and Technology 6 th Floor, 1483 Douglas Street Victoria, British Columbia Canada V8W 3K4 Telephone: Fax: Web: Ordering Marketing Department Open Learning Agency 4355 Mathissi Place Burnaby, British Columbia Canada V5G 4S8 Telephone: or Fax: catalogue@ola.bc.ca Web: ABE Intermediate Level Mathematics Order No. ISBN Module 1: Arithmetic and Estimation VA Module 2: Measurement VA Module 3: Perimeter, Area and Volume VA Module 4: Ratio and Proportion VA Module 5: Percent VA Module 6: Geometry: Introduction VA Module 7: Geometry: Construction VA X Module 8: Statistics VA Module 9: Signed (Rational) Numbers VA Module 10: Algebra: Equations VA X Module 11: Algebra: Polynomials VA Module 12: Powers, Roots and Scientific VA Notation Module 13: Graphing VA Module 14: Trigonometry VA Instructor Package VA Copyright 1999 Province of British Columbia Ministry of Advanced Education, Training and Technology This document may not be reproduced in any form without permission of the Centre for Curriculum, Transfer and Technology. 2

3 Adult Basic Education Intermediate Level Mathematics Module 14 Trigonometry Prepared by Sarah Brodie, North Island College Pat Corbett, North Island College Paul Grinder, Okanagan University College Peter Robbins, Kwantlen University College Ada Sarsiat, Northwest Community College for the Province of British Columbia Ministry of Advanced Education, Training and Technology and the Centre for Curriculum, Transfer and Technology 1999 Revised

4 Contents Learning outcomes... 5 Introduction The right triangle... 7 Exercise Angles and sides... 9 Exercise Pythagorean Theorem Exercise The tangent ratio Exercise Using the tangent ratio Exercise The sine and cosine ratios Exercise Solving triangles Exercise Practice Test Problem Set A Problem Set B Appendix A: Table of Trigonometric Ratios Appendix B: Glossary Answers

5 Learning outcomes Trigonometry is an application of ratio and proportion, geometry, algebra, and measurement. Though it is usually considered an advanced math topic, it is applied in various levels of trades and technology. When you have completed this module, you should be able to: name the parts of a right triangle find the missing side of a right triangle using the Pythagorean Theorem find the measure of an unknown side or angle of a right triangle using sine, cosine or tangent ratios solve problems using right triangle trigonometry Procedure for independent study You will require a scientific calculator, ruler, and protractor. 1. Study each of the units in order, complete all of the exercises, and check your answers. If you need assistance, contact your instructor. 2. Review the definitions and terminology in the Glossary (p 42). 3. For more practice on application problems, complete Problem Sets A & B (p 38 40). 4. Complete the Practice Test (p 36) and check your answers. 5. Complete the MATH 052 unit 5 test. 5

6 Introduction Trigonometry means triangle measure. It is a branch of mathematics which was originally developed to calculate unknown sides and angles in triangles. Trigonometry is very useful in applications such as surveying, navigation, and electronics. Example 1 Suppose you wanted to know the height of a tree without having to climb the tree with a measuring tape. You can find the height by taking two measurements from the ground (Figure 1). Figure 1 Once the distance from the base of the tree is known, as well as the angular measurement to the top of the tree, you can make a scale drawing of a similar triangle and measure the length of the corresponding side (Figure 2). Using the proportion: x 30 m = 35.8 cm 30 cm the tree is 35.8 m tall. Figure 2 By using trigonometry, you can solve problems like this in an even simpler fashion. 6

7 5.1 The right triangle The right triangle in Figure 3 is labelled using standard lettering. The three angles of the triangle are A, B and C. The right angle is C = 90. The sides of the triangle are a, b and c. The line segments AB, AC and BC also denote the sides. Side c is called the hypotenuse of the triangle while sides a and b are called the legs of the triangle. The hypotenuse is always the longest side of the triangle. The Pythagorean Theorem states that for any right triangle, c 2 = a 2 + b 2 Figure 3 Angles of triangles are designated with capital letters. Sides are designated with lower case letters. In a right triangle (Figure 4), sides are related to angles as follows: side a is opposite A side b is opposite B side c is opposite C In a right triangle, the hypotenuse is always opposite the right angle. Note that the names of angles A and B could be reversed and the same for sides a and b. Only side c and angle C are fixed. Notice that the side opposite an angle is not part of the angle side a is adjacent to B side b is adjacent to A Figure 4 Sides that are adjacent to angles are actually part of the angle. The hypotenuse is already named and is not considered to be an adjacent side of an angle. Also, we do not consider the right angle to have adjacent sides. Now complete Exercise 5.1 and check your answers. You may have to sketch triangles and take measurements with a ruler and protractor to answer some questions. 7

8 Exercise All triangles have three sides and three angles. a. true b. false 2. The sum of the measures of the three angles of any triangle is equal to 180. a. true b. false 3. The sum of the length of any two sides of a triangle is always longer than the third side. a. true b. false 4. The side opposite the smallest angle of a triangle has the shortest length. The side opposite the largest angle has the longest length. a. true b. false 5. The perimeter of a triangle is equal to the product of ½ the base times the height. a. true b. false 6. The area of a triangle is equal to the product of ½ the base times the height. a. true b. false 7. All right triangles have one right angle. a. true b. false 8. The side opposite the right angle is called the hypotenuse. a. true b. false 9. The sides that form the right angle are perpendicular to each other. a. true b. false 10. The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. a. true b. false Check your answers with the answer key, located at the end of this unit. 8

9 5.2 Angles and sides The sum of the interior angles of any triangle is 180. In any right triangle (Figure 5), one of the angles is 90. The sum of the other two angles is also 90 or A + B = 90 Example 1 Figure 5 Find B of the triangle in Figure 6. Solution Using the formula: A + B = B = 90 B = Figure 6 B = 62 It is very important that you always know precisely which angles and sides are involved in your calculations. Now complete Exercise 5.2 and check your answers. 9

10 Exercise In the triangle shown, indicate: B a. which side is the hypotenuse b. which side is opposite B a c c. which side is adjacent to A d. which side is adjacent to B C b 40 o A e. the measure of B 2. In PQR, indicate: a. the hypotenuse R p b. the side opposite Q Q c. the side adjacent to P d. the side adjacent to Q q r e. the side opposite P f. the side opposite R P g. if P = 20 then Q = 10

11 3. Determine x in each of the following triangles. 4. In the right triangle shown, identify the large (L), middle (M) and small (S) angles. Also identify the long (l), middle (m) and short (s) sides. Then answer the following questions: a. Which angle is the long side opposite? b. Which angle is the middle side opposite? c. Which angle is the short side opposite? d. Are the above statements true for any triangle? Check your answers with the answer key at the end of this unit. 11

12 5.3 The Pythagorean Theorem The Pythagorean Theorem is probably the most famous theorem in mathematics. It was known to the ancient Chinese, Egyptians and Greeks thousands of years ago. Pythagoras Theorem (Figure 7) states that the square of the hypotenuse of a right triangle equals the sum of the square of its legs or: b c c 2 = a 2 + b 2 This theorem allows us to find the length of the third side of any right triangle when we know the length of two of the other sides. a Figure 7 Example 1 Find the hypotenuse of the triangle in Figure 8. Round your answer to one decimal place. Solution Using the formula c 2 = a 2 + b 2 c 2 = c 2 = c 2 = c = Figure 8 c =

13 Example 2 Find the missing side of the right triangle in Figure 9. Round your answer to one decimal place. Solution The missing side of the right triangle is a leg. It can be named a or b. For this example, you can name it b. c 2 = a 2 + b = b = b = b = b 2 b = b = Figure 9 b 6.5 Note that many of the answers that we will calculate will be irrational numbers (decimal numbers that do not terminate and do not repeat in a predictable way). These numbers are approximate numbers, which can be rounded to a specified number of decimal places. Now complete Exercise 5.3 and check your answers. 13

14 Exercise Name the right angle in each of the triangles shown below. a. b. c. d. C B N O A Z X A B C Y 2. Name the hypotenuse in each triangle shown below. M 3. Find the missing side, x, in each triangle shown below. Round your answers to one decimal place. 14

15 4. The length of a rectangle is 52 cm and the width is 40 cm. Find the length of the diagonal. Round your answer to one decimal place. 15

16 5. A 4 m ladder leans against a building. If the bottom of the ladder is 1.3 m from the bottom of the building, how far up the wall will the ladder reach? Round your answer to one decimal place. 6. A warden observes a fire 4.8 km directly east of his cabin. The nearest water bomber is stationed 10.5 km directly south of the cabin. How far is the water bomber from the fire? Round your answer to one decimal place. 7. A basketball player is 2.3 m tall. He has to make a phone call from a pay phone booth which is 1 m wide, 1 m long and 2 m high. Can he fit himself into the phone booth without sitting down or bending? (Hint: What is the distance between opposite corners of the phone booth?) Answer key at the end of this unit. 16

17 5.4 The tangent ratio The three right triangles in Figure 10 are called similar triangles. They are similar triangles because their corresponding angles are equal. Notice that their sides are not equal. With a metric ruler, measure the opposite and adjacent sides (to the 35 angle) in each triangle and record your results in the table. Round your answers to 0.1 cm. TRIANGLE OPPOSITE SIDE ADJACENT SIDE RATIO OF OPPOSITE ADJACENT CM 4.0 CM DECIMAL EQUIVALENT Figure 10 The table should show that the ratio of opposite side to adjacent side is the same for all three triangles with a 35 angle, regardless of the size of the triangle. This is the core idea of trigonometry. 17

18 For any acute angle of a right triangle, the ratio of the opposite side to the adjacent side (Figure 11) is called the tangent ratio, abbreviated tan. For any right triangle: oppositeside tan A or tan A = adjacent side From your investigation of the three triangles in Figure 10, you can now write: tan 35 = = 0.7 Figure 11 You can use a scientific calculator to determine the value of tan 35 by pressing 35 TAN or TAN 35 depending on your particular make of calculator. Make sure your calculator is in degree mode (DEG) first. You should get approximately (to nearest millionth). Example 1 Use a metric ruler to determine tan 60 in Figure 12. Also determine tan 60 using your calculator. Solution Measure the opposite and adjacent sides of the triangle, then: tan 60 = =1.7 The calculator will show for tan 60. Again, note that tangent values to the nearest millionth in your calculator are much more accurate than the calculated values based on measurements to the nearest tenth. Figure 12 Now complete Exercise 5.4 and check your answers. 18

19 Exercise Use a ruler, then a calculator to find the tangents of the indicated angles. Record your results in the table provided. cm 10 o cm 20 o 45 o cm cm cm cm cm cm 50 o 75 o cm cm Angle length of opposite side decimal equivalent calculator value of tan A length of adjacent side to six decimal places to six decimal places Use a calculator to find the following. Round your answers to four decimal places. a. tan 0 b. tan 1 c. tan 5 d. tan 15 e. tan 22 f. tan 31.4 g. tan 45 h. tan i. tan 64 j. tan 0.05 k. tan 86 l. tan 89 19

20 3. Measure the legs of the triangle to find the following. Round your answers to one decimal place. a. tan 25 b. tan Use a calculator to find the following. Round your answers to three decimal places. a. tan 88 b. tan 89 c. tan 89.9 d. tan These values are very large. When a right triangle has a very large acute angle, the opposite side is much larger e. tan 90 than the adjacent side. 5. Find tan 90. Explain why tan 90 is undefined. Answer key at the end of this unit. 20

21 5.5 Using the tangent ratio If one acute angle and one leg of a right triangle are known, the tangent ratio can be used to find the other leg of the triangle. Example 1 Find side b in the triangle in Figure 13. Round your answer to one decimal place. Solution Side b is opposite 40 and the known side is adjacent to 40, so we use the tangent ratio. tan 40 = oppositeside adjacent side b 13 Rewrite this as a proportion and solve by crossmultiplying. tan 40 1 = b 13 Figure 13 1 b = 13 tan 40 or 1 b = (tan 40 ) 13 b To find 13 tan 40, depending on the type of calculator, press 13 TAN 40 = or TAN = Example 2 Find side a of the triangle in Figure 14. Round your answer to one decimal place. Solution Side a is adjacent to 65 and the known side is opposite 65, so we use the tangent ratio. tan 65 = oppositeside adjacent side 24 a Figure 14 21

22 Rewrite and solve as a proportion: tan = a (tan 65 ) a =1 24 cross - multiply 24 a = tan 65 divide by tan 65 a 11.2 To find 24, depending on the type of calculator, tan 65 press 24 TAN 65 = or TAN = You can also use the tangent ratio to find an acute angle of the right triangle when both legs are known. Example 3 Find angle A of the right triangle in Figure 15. Round your answer to the nearest degree. Solution The known sides are opposite and adjacent to angle A, so we use the tangent ratio. oppositeside 16 tan A = adjacent side 13 To find angle A, with the calculator displaying its tangent of , press INV TAN or 2ndF TAN or SHIFT TAN or TAN 1, to display the angle or Figure 15 When the tangent of an angle is known, the angle is called the inverse tangent or arc tangent. Now complete Exercise 5.5 and check your answers. 22

23 Exercise Find side x in each of the following triangles. Round your answer to one decimal place. oppositeside Remember: tan A =. adjacent side 23

24 2. a. In ABC, C = 90, B = 55 and AC = 18. Sketch and label ABC, then find BC. b. In ABC, C = 90, A = 12 and AC = 43. Sketch and label ABC, then find BC. 3. Find angle x in each of the following triangles. Round answers to one decimal place. oppositeside Remember: tan x =. adjacent side 4. In ABC, C = 90, AC = 5.3 and BC = 7.5. Sketch and label ABC, then find A and B. 24

25 5. Find the missing angles and sides in ABC. Hints: Use A + B = 90 to find B. Use the tangent ratio to find AC. Use the Pythagorean Theorem to find AB. 6. Solve ABC. (find A, ACand AB ). 7. A road rises 6 m vertically for every 50 m of horizontal distance. At what angle is the road inclined? 8. Joan stands 20 m from the base of a large tree. Her line of sight to the top of the tree makes an angle of 54 with the horizontal. Joan is 1.6 m tall. How tall is the tree? 9. A roof rises 1.2 m for every 2 m run. Find the pitch of the roof in degrees (angle x). Answer key at the end of this unit. 25

26 5.6 The sine and cosine ratios The tangent ratio can be used to find missing sides of right triangles. If the hypotenuse of a right triangle is unknown, there are two other trigonometric ratios that are useful: the sine ratio, abbreviated sin, pronounced sign the cosine ratio, abbreviated cos, pronounced kose For any right triangle (Figure 16): sin A or sin A = oppositeside hypotenuse Example 1 cos A or cos adjacent side A = hypotenuse Figure 16 Find side AB in ABC (Figure 17). Solution An angle and its opposite side are known. To find the hypotenuse, use the sine ratio. sin 41 = 8 AB Rewrite and solve as a proportion: sin 41 8 = 1 AB (sin 41 ) AB= 8 8 AB= sin 41 Figure 17 To find 8 on a calculator, sin 41 press 8 SIN 41 = or 8 41 SIN = 26

27 Example 2 Find side x of the triangle in Figure 18. Solution Side x is adjacent to the 68 angle and the hypotenuse is 9.6 cm. The cosine ratio can be used to find side x. Figure 18 x cos 68 = 9.6 cos 68 x = x = 9.6 cos 68 = cm To find 9.6 cos 68 on a calculator, press 9.6 COS 68 = or COS = Example 3 Find A of ABC when AB = 44, AC = 29 and C = 90. Solution Sketch ABC and label the given parts (Figure 19). The hypotenuse and adjacent side to A are given, so use the cosine ratio to find A. cos A = = A = 48.8 Figure 19 To find A, with the calculator displaying its cosine of 0.659, press INV COS or 2ndF COS or COS 1, to display the angle When the cosine of an angle is known, the angle is called the inverse cosine or arc cosine. A common mnemonic (memory assisting device) to remember the sine, cosine, and tangent ratios is the word SOHCAHTOA (pronounced sew caw toa ), which is shorthand for the ratios Sine = Opposite, Hypotenuse Cosine = Adjacent, Hypotenuse Opposite Tangent = Adjacent Now complete Exercise 5.6 and check your answers. 27

28 Exercise Use a calculator to find the following pairs of sine and cosine values. Round answers to four decimal places. sin A cos B a) sin 1 cos 89 b) sin 5 cos 85 c) sin 10 cos 80 d) sin 22.5 cos 67.5 e) sin 30 cos 60 f) sin 45 cos 45 g) sin 60 cos 30 h) sin 67.5 cos 22.5 i) sin 80 cos 10 j) sin 85 cos 5 k) sin 89 cos 1 Study both columns of answers. Notice that for each pair of values, A + B = 90 ( complementary angles), and sin A = cos B. For any pair of complementary angles, the sine of one angle equals the cosine of the other, or in other words, sin A = cos (90 A). 2. From the triangle shown, find the following to one decimal place. a. sin 37 b. cos 53 c. sin 53 d. cos 37 e. suppose that sin 20 = What is cos 70? 4 37 o 5 53 o 3 3. Find side x in each triangle. Round your answers to one decimal place. For each of the following triangles, ask yourself the following three questions: 1. Name side x. Is it the hypotenuse (H), the opposite side (O), or the adjacent side (A)? 2. Name the given side. Is it the hypotenuse, the opposite side, or the adjacent side? 3. Using SOHCAHTOA, which ratio should be used? sin, cos, or tan? 28

29 a. Name side x. Is it the hypotenuse, b. Name side x. H, O or A? opposite, or adjacent to 30? Name side 13. H, O or A? Name the given side 40. Is it Use sin, cos or tan? the H, O or A side? Which ratio? sin, cos or tan? X o 25 o X Ask and answer the above three questions for each triangle below. c. d. 60 o X X 70 o e. f. X 17 o X o 6.5 g. h o X 45 o X In ABC C = 90, B = 24 and AC = 75 cm. Find AB. 5. In ABC C = 90, A = 69 and AB = 24.7 cm. Find BC. 29

30 6. Find x in each of the following. Round your answers to the nearest degree. (use SOHCAHTOA) a. b X c. d. X 305 X X e. f X X 7. In ABC C = 90, AC = 14 and AB = 19. Find A and B to the nearest tenth of a degree. 8. A 10 m long ramp rises to a point 1 m above the ground. Find the angle of inclination of the ramp. 1m 10m X 30

31 9. An eagle is sitting on a tree 14 m above the ground. The eagle sights a mouse on the ground at an angle of 58 with the tree. How far is the eagle from the mouse? 58 o 14 m 10. An airplane travels at 450 km/hr in a N50 E direction. How far east has the plane travelled after 1 hour? N 450 km 50 o W E S Answer key at the end of this unit. 31

32 5.7 Solving triangles To solve a triangle means to determine all the unknown sides and angles. Before this is possible, you have to know three parts of the triangle. One part must be a side.. Example 1 Solve ABC where C = 90, A = 53 and c = 18. B Solution Sketch ABC and label the known parts (Figure 20), then find B. a c = 18 A + B = B = 90 B = B = 37 C b Figure o A Find side a. Side a is opposite 53 and c = 18 is the hypotenuse. Using the sine ratio, sin 53 = a sin 53 = a a 14.4 There are several ways of determining side b. The safest method, which involves only given information, is to use the cosine ratio, since side b is adjacent to 53. cos 53 = b cos 53 = b b 10.8 Both the Pythagorean Theorem and the tangent ratio require calculated answers to find side b. Even though they are valid methods, they are not the most reliable. You could use a formula such as: cos 37 = a to find side a. 18 This requires that you involve the calculated value of 37. If 37 is a wrong answer, then side a will also be a wrong answer. It is always more reliable to involve given information in a calculation whenever possible. Also note that if you round a value that you use in a subsequent calculation, the next value that you find will also be an approximation. 32

33 Example 2 Solve the triangle in Figure 21. Solution Find the hypotenuse, RT as follows: RT = RT = = 325 RT = Figure 21 Find R as follows; side 15 is opposite R, side 10 is adjacent to R, so use the tangent ratio, 15 tan R = = R 56.3 You can use the tangent ratio to find T: 10 tan T = 15 T 33.7 Although it is not as reliable, it is also just as easy to use: T + R = 90 T = 90 T = = 33.7 Now complete Exercise 5.7 and check your answers. 33

34 Exercise Solve the following triangles. Find the angles to the nearest degree and the sides to one decimal place. a. b. A C a = 25 B 40 o b c b c = 28 A C a = 5 B c. d. b A C B c = o a B c a = 35 A b = 40 C e. f. A A b 23 o C c c b = 7.6 a = o C B B a 2. Solve ABC where C = 90, A = 16 and b = Solve ABC where C = 90, a = 12 and c = Solve ABC where C = 90, a = 1 and b = 1. 34

35 5. Solve ABC where C = 90, A = 40 and B = In ABC, note that ACB = 90, A = 60 and altitude CD = 12. Find the following: a. B b. AC c. BC d. AB 7. In ABC, ACB = 90, A = 65 and AB = 20. Find altitude CD. 8. An airplane flies 300 km at a heading of 80. How far north and how far east has it travelled? Answer key at the end of this unit. Now complete the Practice Test and check your answers. 35

36 Practice Test 1. Use a calculator to find the following. Round your answers to four decimal places. a. sin 36 b. cos 9.2 c. tan 87 d. sin Use a calculator to find A in each of the following. Round your answers to the nearest degree. a. cos A = b. tan A = c. sin A = 11 d. tan A = 3 3. Find side x or angle x in each of the following. Round your answers to one decimal place. Remember SOHCAHTOA. 36

37 4. Solve ABC where C = 90, B = 40 and a = Solve ABC where C = 90, a = 24 and c = Find the altitude CD of ABC if A = 25 and AB= 1.2. Note that ABC is a right triangle. C B D A 7. Engineers measure the right-angled distances beside a mountain as shown. They propose to build a tunnel from A to C through the mountain. What is the length of the tunnel? 8. Grace wants to know how far it is to the other side of the river. She makes the measurements as shown. What is the distance from A to B across the river? A 35 o C 20 m B Answer key at the end of this unit. 37

38 Problem Set A Read each question carefully, then draw a diagram. When you have finished the diagram, read the question again to make sure you have interpreted it properly, then work out the solution. 1. Pat wants to know the height of a tree in her yard. She observes that from a distance of 20 m from the tree, the angle of elevation to the top of the tree is 32. What is the height of the tree to the nearest tenth of a metre? 2. From a lighthouse 16 m above sea level, the angle of depression to a small boat is 12. How far from the foot of the lighthouse is the boat to the nearest metre? 3. A kite is 40 m high when 210 m of string is used. What angle does the kite string make with the horizontal to the nearest degree? 4. Marianne wants to know how far it is across a river. She notices a tree at point X straight across from point Y. She walks 10 m along the river bank to point Z and observes that the angle to the tree is 65. What is the distance across the river from point X to point Y to the nearest hundredth of a metre? X River Y 10 m 65 o Z 5. A plot of land has the shape of a right triangle. The longest side is 37 m and lies at an angle of 53 to the shortest side. Find the area of the plot to the nearest square metre. 38

39 6. A painter is using a ladder to paint a high factory wall. The ladder is 5 m long and for safety, must never be used at an angle to the vertical of less than 15 or more than 40. If he can reach 1 m above the top of the ladder, what is the maximum height to the nearest tenth of a metre that he can paint? (Hint: what angle would give the painter the greatest height?) 7. Jack is building a garden shed of his own design. It has a simple sloping roof. The two walls on which the roof will rest are 3.2 m apart and one wall is 0.5 m higher than the other. Allowing 0.25 m for overhang at either end, how long (to the nearest hundredth) will the roof beams have to be? What will the slope of the roof be to the nearest degree with respect to the horizontal? Sloping roof Shed 8. Joan is welding a piece of modern sculpture. Part of the design includes an A-frame structure. Joan wants the two thin bars that make up the sides of the frame to form an angle of 54 at the top and she wants the frame to be 2.2 m high. How long will each bar have to be to the nearest thousandth? (Hint: you need a right triangle to use a trigonometric ratio.) 9. A new ski-lift is being built at the slopes. The base of the lift is at an elevation of m, but the elevation of the top station is not accurately known. A survey of the site shows the base and the top station are m apart in horizontal distance and a line of sight to the top station angles up at 38. Find the length of steel cable (to the nearest 10 m) that will be needed for the endless loop on which the chairs will hang. Allow for an additional 5% of the total length for sags, joining, etc. 10. A mathematical spider spinning a web across an alleyway between two buildings manages to stretch a length of silken thread across from the top of a 2 m high doorway to a point on the wall of the opposite building 1 m above the ground. The spider just happened to make the angle of the thread exactly 25 to the horizontal. How far apart are the buildings to the nearest centimetre? 11. The roof of a small pup tent is made of a rectangular piece of material. If the tent is to be 2.2 m long, the roof sloping up at 48 to the horizontal and with the poles 1.4 m high, how many square metres of material will be needed to make the tent roof? (nearest hundredth) 39

40 Problem Set B 1. Given that the property line is halfway between the two houses on the plan below, what is the distance between the two houses to the nearest tenth of a metre? Property line House A House B 19.4 m 72 o Survey post Road 2. How far is the corner of house B from the survey post to the nearest tenth of a metre? 3. A carpenter is instructed to cut a right-angled wooden wedge 25 cm long in the base with an angle of 12. How long will the sloping surface of the wedge be to the nearest millimetre? 4. From a ladder, Wayne looks at a building 35 m away. He notes that the angle of elevation to the top of the building is 19 and the angle of depression to the bottom of the building is 7. How high is the building? 5. A slide in the Water Park is 9 m high. If the actual length of the slide is 14 m, what angle does the slide make with the horizontal 40

41 Appendix A: Table of Trigonometric Ratios Angle sin cos tan Angle sin cos tan NOTE: values are correct to the nearest 1/1000th 41

42 Appendix B: Glossary Acute angle Any angle whose measure is greater than 0 and less than 90. A right triangle always has two acute angles and one right angle. In Figure 22, A and B are acute angles. Cosine of an angle B For a specified acute angle of a right triangle, the cosine of the angle is equal to the ratio of the length of the adjacent leg to the length of the hypotenuse. For the triangle in Figure 22: c a b cos A = c Hypotenuse A b Figure 22 C The longest side of a triangle, opposite the right angle. In Figure 22, side c is the hypotenuse. Leg Either of the two sides of a right triangle that are opposite the two acute angles. In Figure 22, sides b and a are the legs. Pythagorean Theorem The Pythagorean Theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the length of the legs. For the triangle in Figure 22: c 2 = a 2 + b 2 Right triangle A triangle that has a 90 angle. Figure 22 is a right triangle. Sine of an angle For a specified angle of a right angle triangle, the sine of the angle is equal to the ratio of the length of the opposite leg to the length of the hypotenuse. For the triangle in Figure 22: a sin A = c Tangent of an angle For a specified angle in a right angle triangle, the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. For the triangle in Figure 22: a tan A = b 42

43 Answers Exercise page 8 1. T 2. T 3. T 4. T 5. F 6. T 7. T 8. T 9. T 10. T Exercise pages a. c or AB b. b or AC c. b or AC d. a or BC e a. PQ or r b. PR or q c. PR or q d. QR or p e. QR or p f. PQ or r g a. 60 b. 80 c. 15 d. 90 e. 36 f a. large or L b. middle or M c. short or S d. yes Exercise pages a. C b. O c. B d. Y 2. a. c b. AB c. PR d. e 3. a b c d e f. 1.1 g h cm m km 7. Yes, the distance between opposite corners is more than 2.4 m. Exercise pages calculator answers to 6 decimal places: tan 10 = tan 20 = , tan 45 = tan 50 = tan 75 = a b c d e f g h i j k l a. tan 25 = 0.5 b. tan 65 = tan 88 = tan 89 = tan 89.9 = tan = tan 90 undefined (your calculator should display ERROR) 5. tan 90 = undefined. Since it is impossible to draw a triangle with two 90 angles, tan 90 cannot be calculated. 43

44 Exercise pages a. tan 15 = x / 10, x = 2.7 b. tan 30 = 8 / x, x = 13.9 c. tan 65 = x / 19, x = 40.7 d. tan 40 = x / 17.3, x = 14.5 e. tan 53 = 239 / x, x = f. tan 39 = x / 10.9, x = 8.8 g. tan 80 = 982 / x, x = h. tan 13 = x / 50.7, x = a. BC = 12.6 b. BC = a b. 54 c d A = 54.8 B = B = 18 AC = 4.5 AB = A = 40 AC = 88.2 AB = m m = 29.1 m Exercise pages sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos Notice that both answer columns are the same. Also notice that sin 1 = cos 89 and that = 90, sin 5 = cos 85 and = 90 and so on down the columns. 2. a. sin 37 = 3/5 = 0.6 b. cos 53 = 3/5 = 0.6 c. sin 53 = 4/5 = 0.8 d. cos 37 = 4/5 = 0.8 Notice that sine 37 = cos 53 since = 90. Also sin 53 = cos 37 because = 90. e. cos 70 = since = a. sin 30 = x / 40, x = 20 b. cos 25 = x / 13, x = 11.8 c. tan 60 = x / 9, x = 15.6 d. sin 70 = 142 / x,x = e. cos 17 = 91 / x, x = 95.2 f. sin 53 = x / 6.5, x = 5.2 g. cos 45 = 15.9 / x, x = 22.5 h. sin 72.5 = 3.6 / x, x = cm cm 6. a. sin x = 15 / 20, x = 49 b. cos x = 17.3 / 30, x = 55 c. cos x= 3.6 / 5.1, x = 45 d. sin x = 305 / 462, x = 41 e. cos x = 4.85 / 6.02, x = 36 f. cos x = 83.4 / 9.5, x = A = 42.5 and B = m km 44

45 Exercise pages a. B = 50, b = 6, c = 7.8 b. A = 63, B = 27, b = 12.6 c. A = 25, a = 6.8, b = 14.5 d. c = 53.2, A = 41, B = 49 e. A = 48, b = 54, c = 80.7 f. B = 67, a = 3.2, c = B = 74, a = 26.4, c = b = 5, A = 67, B = c = 1.4, A = 45, B = There are an infinite number of triangles. 6. B = 30, AC = 13.9, BC = 24, AB = BC = 20 sin 65 = and B = 25, therefore CD = km north and km east Practice Test - pages a b c d a. 9 b. 58 c. 33 d a b. 6 c d e f A = 50, b = 73.8, c = b = 18, A = 53, B = cos 25 = km m AC CD so AC= Now sin 25 = and CD = Problem Set A pages m m m m m 7. roof beams are 3.73 m long, slope is m m cm m 2 Problem Set B page m m mm m

46 Notes